Two-dimensional solitary water waves with constant vorticity, Part II: the deep capillary case

2024/08/06 by James D. Rowan, Rowan, James, Lizhe Wan +1
Earth and Planetary Sciences · Environmental Science · #35Q35 #76B15 #Analysis of PDEs (math.AP) #Coastal and Marine Dynamics #FOS: Mathematics #Methane Hydrates and Related Phenomena #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.2408.03428

openalex publication_date 2024/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the two-dimensional capillary water waves with nonzero constant vorticity in infinite depth. We first derive the Babenko equation that describes the profile of the solitary wave. When the velocity c is close to a critical velocity and a sign condition involving the physical parameters is met, the Babenko equation can be reduced to the stationary focusing cubic nonlinear Schrödinger equation plus perturbative error. We show the existence of a critical value of a dimensionless physical parameter below which at least two families of velocities satisfy the focusing condition and above which only one does. This gives the existence of small-amplitude solitary wave solutions for the water wave system with constant vorticity.

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